mathematics in the modern world - mathematics in our world · mathematics in the modern world the...

Mathematics in the Modern World

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Mathematics in the Modern WorldMathematics in Our World

Joel Reyes Noche, [email protected]

Department of MathematicsAteneo de Naga University

Council of Deans and Department Chairs of Colleges of Arts and Sciences Region VConference and Enrichment Sessions on the New General Education Curriculum

Ateneo de Naga University, Naga CitySeptember 2, 2017–September 3, 2017


The Nature of Mathematics Mathematics in Our World 2/35

Mathematics in Our WorldMathematics is a useful way to think about nature and our world

Learning outcomes

I Identify patterns in nature and regularities in the world.

I Articulate the importance of mathematics in one’s life.

I Argue about the nature of mathematics, what it is, how it isexpressed, represented, and used.

I Express appreciation for mathematics as a human endeavor.



Nature by NumbersA short movie by Cristobal Vila, 2010

Original (3:44): https://vimeo.com/9953368

Alternative soundtrack (4:04): https://vimeo.com/29379521

https://vimeo.com/9953368

https://vimeo.com/29379521



Fibonacci sequence(Vila, 2016)

The Fibonacci sequence “is an infinite sequence of naturalnumbers where the first value is 0, the next is 1 and, from there,each amount is obtained by adding the previous two.”



Fibonacci spiral(Vila, 2016)

Circular arcs connect the opposite corners of squares in theFibonacci tiling.



Golden spiral(Golden spiral in rectangles, 2008)

r = ϕ2θ/π where θ is in radians and ϕ = 1+√5

2 is the golden ratio



Comparison of spirals(Approximate and true Golden Spirals, 2009)

Quarter-circles in green, golden spiral in red, overlaps in yellow



Nautilus spiral(Vila, 2016)



Vila’s mistake(Vila, 2016)

Vila admits he made a mistake in the animation for the Nautilusshell. (It is neither a Fibonacci spiral nor a golden spiral.)



Golden rectangle(Vila, 2016)



Golden ratio(Vila, 2016)



Golden angle(Vila, 2016)



Golden angle and arrangement of sunflower seeds(Vila, 2016)



Fibonacci numbers and arrangement of sunflower seeds(Vila, 2016)



Fibonacci numbers and arrangement of sunflower seeds(continuation) (Vila, 2016)



Delaunay triangulation and Voronoi diagram(Vila, 2016)



Delaunay triangulation and Voronoi diagram(continuation) (Vila, 2016)



Delaunay condition(Vila, 2016; Delaunay triangulation, 2017)

A Delaunay triangulation for a set of points in a plane is a triangulationsuch that no point is inside the circumcircle of any triangle.



Dragonfly wings(Vila, 2016)



Relationship between golden ratio and Fibonacci sequence(Vila, 2016; Golden ratio, 2017)

F0 = 0

F1 = 1

Fn = Fn−1 + Fn−2 for n > 1, n ∈ Z

Fn =ϕn − (1− ϕ)n√

5=ϕn − (−ϕ)−n√

5

limn→∞

Fn+1

Fn= ϕ



Sample exam questions

1. Let ϕ = 12

(1 +√

5). Show that ϕ = ϕ2 − 1 =

1

ϕ+ 1.

2. Let φ = 12

(1−√

5). Show that φ = φ2 − 1 =

1

φ+ 1.

3. Explain why you think the golden ratio is defined to beϕ = 1

2

(1 +√

5)

and not φ = 12

(1−√

5).



Mathematics is a useful way to think about nature(Stewart, 1995, p. 19)

Whatever the reasons, mathematics definitely is auseful way to think about nature. What do we want it totell us about the patterns we observe? There are manyanswers. We want to understand how they happen; tounderstand why they happen, which is different; toorganize the underlying patterns and regularities in themost satisfying way; to predict how nature will behave; tocontrol nature for our own ends; and to make practicaluse of what we have learned about our world.Mathematics helps us to do all these things, and often itis indispensable.



Sample reading assignment

Read Stewart (1995) and be ready to answer the followingdiscussion questions.

I Which sentence or paragraph in the book is your favorite?Why?

I Is there any statement or point of view in the book that youdisagree with?

I How would you summarize each of the nine chapters (in one,two, or three sentences per chapter)?

I How does Stewart differentiate the external aspects ofmathematics from the internal aspects of mathematics?

I What term does Stewart use to describe his dream of aneffective mathematical theory of form and the emergence ofpattern?



Sample assignment

Using Stewart (1995) as your reference, write an essay of around250 to 350 words answering exactly one of the following questions.

I How did Stewart explain why numbers from the Fibonacciseries appear when some features in plants are counted?

I Leonhard Euler got into an argument with Daniel Bernoullibecause their solutions to the one-dimensional wave equationdiffered. How did Stewart explain the argument’s resolution?

I How did Stewart use Poincare’s concept of a phase space toexplain why tides are predictable but weather is not?

I How did Stewart use coupled oscillator networks to modelanimal gaits?



Is math discovered or invented?A TED-Ed Original lesson by Jeff Dekofsky, 2014

(5:11): https://youtu.be/X xR5Kes4Rs

See also http://ed.ted.com/lessons/

is-math-discovered-or-invented-jeff-dekofsky.

https://youtu.be/X_xR5Kes4Rs

http://ed.ted.com/lessons/is-math-discovered-or-invented-jeff-dekofsky

http://ed.ted.com/lessons/is-math-discovered-or-invented-jeff-dekofsky



Philosophy of mathematics(Philosophy of Mathematics, 2017)

“Mathematical realism [...] holds that mathematical entities existindependently of the human mind. Thus humans do not inventmathematics, but rather discover it [...].” One form ofmathematical realism is Platonism.

“Mathematical anti-realism generally holds that mathematicalstatements have truth-values, but that they do not do so bycorresponding to a special realm of immaterial or non-empiricalentities.” One form of mathematical anti-realism is formalism.



Platonism

“Mathematical Platonism is the form of realism that suggests thatmathematical entities are abstract, have no spatiotemporal orcausal properties, and are eternal and unchanging.” (Philosophy ofMathematics, 2017)

(Richard Hamming, 2013)

“Very few of us in our saner moments believe thatthe particular postulates that some logicians havedreamed up create the numbers—no, most of usbelieve that the real numbers are simply there andthat it has been an interesting, amusing, and im-portant game to try to find a nice set of postulatesto account for them.” (Hamming, 1980, p. 85)



Formalism

“Formalism holds that mathematical statements may be thoughtof as statements about the consequences of certain stringmanipulation rules.” (Philosophy of Mathematics, 2017)

(David Hilbert, 2017)

“Mathematics, according to David Hilbert(1862-), is a game played according to certainsimple rules with meaningless marks on paper.”(Stabler, 1935, p. 24)



Ultrafinitism

“[C]onstructivism involves the regulative principle that only mathematicalentities which can be explicitly constructed in a certain sense should beadmitted to mathematical discourse. [...]

Finitism is an extreme form of constructivism, according to which amathematical object does not exist unless it can be constructed from naturalnumbers in a finite number of steps. [...]

Ultrafinitism is an even more extreme version of finitism, which rejects not onlyinfinities but finite quantities that cannot feasibly be constructed with availableresources.” (Philosophy of Mathematics, 2017)

(Doron Zeilberger, 2007)

“What is completely meaningless is any kind of infinite,actual or potential. So I deny even the existence of thePeano axiom that every integer has a successor. [...]

The phrase ‘for all positive integers’ is meaningless. [...]

Similarly, Euclid’s statement: ‘There are infinitely manyprimes’ is meaningless.” (Zeilberger, 2001, p. 5)



Constructive and nonconstructive proofs

TheoremThere exist irrational numbers a and b such that ab is rational.

Nonconstructive proof.

Consider√

2√2. If it is rational, then the proof is complete. If it is

not rational, then take a =√

2√2

and b =√

2 so that ab = 2.

Constructive proof (sketch).

Take a =√

2 and b = 2 log2 3.


References 33/35

Approximate and true Golden Spirals. (2009, August 29). In WikimediaCommons, the free media repository. Retrieved April 20, 2017,from https://upload.wikimedia.org/wikipedia/commons/

thumb/a/a5/FakeRealLogSpiral.svg/

640px-FakeRealLogSpiral.svg.png

David Hilbert. (2017, February 14). In Wikimedia Commons, the freemedia repository. Retrieved April 22, 2017, fromhttps://upload.wikimedia.org/wikipedia/commons/7/79/

Hilbert.jpg

Delaunay triangulation. (2017, February 27). In Wikipedia, The FreeEncyclopedia. Retrieved April 20, 2017, fromhttps://en.wikipedia.org/w/index.php?title=Delaunay

triangulation&oldid=767721079

Doron Zeilberger. (2007, December 13). In Wikimedia Commons, thefree media repository. Retrieved April 22, 2017, fromhttps://upload.wikimedia.org/wikipedia/commons/thumb/

e/e9/Doron Zeilberger %28circa 2005%29.jpg/

365px-Doron Zeilberger %28circa 2005%29.jpg

https://upload.wikimedia.org/wikipedia/commons/thumb/a/a5/FakeRealLogSpiral.svg/640px-FakeRealLogSpiral.svg.png



https://upload.wikimedia.org/wikipedia/commons/7/79/Hilbert.jpg

https://upload.wikimedia.org/wikipedia/commons/7/79/Hilbert.jpg

https://en.wikipedia.org/w/index.php?title=Delaunay_triangulation&oldid=767721079

https://en.wikipedia.org/w/index.php?title=Delaunay_triangulation&oldid=767721079

https://upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Doron_Zeilberger_%28circa_2005%29.jpg/365px-Doron_Zeilberger_%28circa_2005%29.jpg




References 34/35

Golden ratio. (2017, March 21). In Wikipedia, The Free Encyclopedia.Retrieved April 20, 2017, from https://en.wikipedia.org/w/

index.php?title=Golden ratio&oldid=771516527

Golden spiral in rectangles. (2008, January 27). In Wikimedia Commons,the free media repository. Retrieved April 19, 2017, fromhttps://upload.wikimedia.org/wikipedia/commons/7/70/

Golden spiral in rectangleflip.png

Hamming, R. W. (1980). The unreasonable effectiveness ofmathematics. American Mathematical Monthly, 87, 81–90.

Philosophy of Mathematics. (2017, February 6). In Wikipedia, The FreeEncyclopedia. Retrieved April 22, 2017, fromhttps://en.wikipedia.org/w/index.php?title=Philosophy

of mathematics&oldid=763957978

Richard Hamming. (2013, August 7). In Wikimedia Commons, the freemedia repository. Retrieved April 22, 2017, fromhttps://upload.wikimedia.org/wikipedia/en/0/08/

Richard Hamming.jpg

https://en.wikipedia.org/w/index.php?title=Golden_ratio&oldid=771516527

https://en.wikipedia.org/w/index.php?title=Golden_ratio&oldid=771516527

https://upload.wikimedia.org/wikipedia/commons/7/70/Golden_spiral_in_rectangleflip.png

https://upload.wikimedia.org/wikipedia/commons/7/70/Golden_spiral_in_rectangleflip.png

https://en.wikipedia.org/w/index.php?title=Philosophy_of_mathematics&oldid=763957978

https://en.wikipedia.org/w/index.php?title=Philosophy_of_mathematics&oldid=763957978

https://upload.wikimedia.org/wikipedia/en/0/08/Richard_Hamming.jpg

https://upload.wikimedia.org/wikipedia/en/0/08/Richard_Hamming.jpg


References 35/35

Stabler, E. R. (1935). Interpretation and comparison of three schools ofthought in the foundations of mathematics. The MathematicsTeacher, 28, 5–35.

Stewart, I. (1995). Nature’s numbers: The unreal reality of mathematics.New York, NY: BasicBooks.

Vila, C. (2016, September). Nature by numbers. The theory behind thismovie. Retrieved April 18, 2017, fromhttp://www.etereaestudios.com/docs html/nbyn htm/

about index.htm

Zeilberger, D. (2001, November 27). “Real” analysis is a degenerate caseof discrete analysis. Retrieved April 22, 2017, fromhttp://www.math.rutgers.edu/~zeilberg/mamarim/

mamarimPDF/real.pdf

http://www.etereaestudios.com/docs_html/nbyn_htm/about_index.htm

http://www.etereaestudios.com/docs_html/nbyn_htm/about_index.htm

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf

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